Rickard Equivalences and Block Theory

Michel Broué

École Normale Supérieure, Paris

Four lectures given at the International Conference on Group theory “Groups 93” Galway, August 1993

1. Introduction

Control of fusion.

Let G be a finite group, and let p be a prime number.

1.1. Definition. We say that a subgroup H of G controls the fusion of p–subgroups of G if the following two conditions are fulfilled : (C1) H contains a Sylow p–subgroup S_{p} of G, (C2) whenever P is a subgroup of S_{p} and g is an element of G such that gP g^{−1} ⊆ S_{p},

there exist z in the centralizer C_{G}(P) of P in G, and h in H, such that g = hz.

1.2. Basic example. We denote by O_{p′}G the largest normal subgroup of G with order prime to p. Then if H is a subgroup of G which “covers the quotient” G/O_{p′}G (i.e., if G = HO_{p′}G), then H controls the fusion of p–subgroups of G.

The following two results provide fundamental examples where the converse is true. The first one is due to Frobenius and was proved in 1905. The second one was proved by Glauberman for the case p = 2 (see [Gl]), and for odd p it is a consequence of the classification of non abelian finite simple groups (see also [Ro] for an approach not using the classification).

1.3. Theorems.

(Fr) Assume that a Sylow p–subgroup S_{p} of G controls the fusion of p–subgroups of

G. Then G = S_{p}O_{p′}G. (Gl) Assume that there exists a p–subgroup P of G whose centralizer C_{G}(P) controls

the fusion of p–subgroups of G. Then G = C_{G}(P)O_{p′}G.

Groups with abelian Sylow p–subgroups.

A classical example where a subgroup controls the fusion of p–subgroups is given by an old result of Burnside :

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20, 20G.

Key words and phrases. Finite Groups, Algebras, Representations.

Assume that the Sylow p–subgroups of G are abelian. Let H be the normalizer of one of them. Then H controls the fusion of p–subgroups of G.

If G is p–solvable, this is once again a particular case of 1.2, since it is not difficult to prove the following result.

1.4. Proposition. Let G be p–solvable and let H be the normalizer of a Sylow p– subgroup. If the Sylow p–subgroups of G are abelian, then G = HO_{p′}G.

The situation may look quite different if G is a non abelian simple group with abelian Sylow p–subgroups. Indeed, in this case, we have G ≠ HO_{p′}G whenever H normalizes a non-trivial p–subgroup of G.

For example, there seems to be an enormous difference between the Monster group, a non abelian simple group of order

2^{46} · 3^{20} · 5^{9} · 7^{6} · 11^{2} · 13^{3} · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 ≃ 8.10^{53} ,

and the normalizer of one of its Sylow 11–subgroups, a group of order 72600, isomorphic to (C_{11} × C_{11}) ⋊ (C_{5} × SL_{2}(5)) (here we denote by C_{m} the cyclic group of order m).

Nevertheless, there still is a strong connection between G and the normalizer of a Sylow p–subgroup, which is a kind of generalization of the “factorization situation” given by theorem 1.3. In order to express this connection, we need to introduce the language of block theory.

The principal block.

Let K be a finite extension of the field of p–adic numbers ℚ_{p} which contains the |G|-th roots of unity. Thus the group algebra KG is a split semi–simple K–algebra. Let O be the ring of integers of K over ℤ_{p}. We denote by 𝔭 the maximal ideal of O, and we set k := O/𝔭. If JkG denotes the Jacobson radical of the group algebra kG, the algebra kG/JkG is a split semi–simple k–algebra.

[figure: K; O; k = O/𝔭; ℚ_{p}; ℤ_{p}; 𝔽_{p} = ℤ_{p}/pℤ_{p}]

The decomposition of the unity element of OG into a sum of orthogonal primitive central idempotents 1 = ∑ e corresponds to the decomposition of the algebra OG into a direct sum of indecomposable two–sided ideals OG = ⨁ B (B = OGe), called the blocks of OG. For B a block of OG, we set KB := K ⊗_{O} B and kB := k ⊗_{O} B.

By reduction modulo 𝔭, a primitive central idempotent remains primitive central, and consequently kG = ⨁ kB is still a decomposition into a direct sum of indecomposable two–sided ideals, called the blocks of kG.

OG = ⨁ B
↓  ↓
kG = ⨁ kB

The augmentation map OG → O factorizes through a unique block of OG called the principal block and denoted by B_{p}(G).

If B is a block of OG, we denote by Irr(B) the set of all isomorphism classes of irreducible representations of the algebra KB. The set Irr(B) will be identified with a subset of the set Irr(G) of characters of irreducible representations of G over K.

For χ ∈ Irr(G), we denote by ker(χ) the kernel of the corresponding representation of G. In other words, we have

ker(χ) = {z ∈ G; (∀g ∈ G)(χ(zg) = χ(g))} .

The factorization G = HO_{p′}G can be interpreted in terms of principal blocks as follows.

1.5. Proposition.

(1) We have ⋂_{χ∈Irr(B_{p}(G))} ker(χ) = O_{p′}G .

(2) If H is a subgroup of G, the following assertions are equivalent

(i) G = HO_{p′}G , (ii) the map Res_{H}^{G} induces a bijection from Irr(B_{p}(G)) onto Irr(B_{p}(H)) .

What happens in the general case (where the Sylow p–subgroups of G are abelian and H is the normalizer of one of them) will first be illustrated by the example of the group G = 𝔄_{5}. The bijection Res_{H}^{G} of the previous proposition is replaced by a “bijection with signs” between Irr(B_{p}(G)) and Irr(B_{p}(H)) .

The case of G = 𝔄_{5}.

Let G be the alternating group on five letters. Then |G| = 2^{2} · 3 · 5, and for all prime number p which divides |G|, the Sylow p–subgroups of G are abelian. Let us examine the principal p–blocks of G and their connections with the corresponding Sylow normalizers.

1.6. Character table of 𝔄_{5}

(1)  (2)  (3)  (5)  (5′)
1  1  1  1  1  1
χ_{4}  4  0  1  −1  −1
χ_{5}  5  1  −1  0  0
χ_{3}  3  −1  0  (1 + \sqrt{5})/2  (1 − \sqrt{5})/2
χ′_{3}  3  −1  0  (1 − \sqrt{5})/2  (1 + \sqrt{5})/2

We have

Irr(B_{2}(G)) = {1 , χ_{5} , χ_{3} , χ′_{3}} ,
Irr(B_{3}(G)) = {1 , χ_{4} , χ_{5}} ,
Irr(B_{5}(G)) = {1 , χ_{4} , χ_{3} , χ′_{3}} , .

For each p ∈ {2, 3, 5}, let us denote by S_{p} a Sylow p–subgroup of G. We shall point out that there exists an isomorphism

I_{p}: ℤIrr(B_{p}(N_{G}(S_{p})))⟶^{∼}ℤIrr(B_{p}(G))

such that :

(I1) I_{p} is an isometry, (I2) it preserves the character degrees modulo p, (I3) it preserves the values on the p-elements. The preceding properties express the fact that I_{p} is an isotypy between B_{p}(G) and

B_{p}(N_{G}(S_{p})), as we shall explain below in §2.

The case p = 2. The normalizer N_{G}(S_{2}) of a Sylow 2–subgroup S_{2} of G is isomorphic to the alter-

nating group 𝔄_{4}. The principal block B_{2}(N_{G}(S_{2})) is its only 2–block. Let us change the sign of certain irreducible characters in its character table.

1.7. Character table of 𝔄_{4}

(1)  (2)  (3)  (3′)
1  1  1  1  1
−α_{3}  −3  1  0  0
−α_{1}  −1  −1  (1 + \sqrt{−3})/2  (1 − \sqrt{−3})/2
−α′_{1}  −1  −1  (1 − \sqrt{−3})/2  (1 + \sqrt{−3})/2

We denote by I_{2} the map

I_{2} : ( \begin{matrix}1 \\ −α_{3} \\ −α_{1} \\ −α′_{1} \end{matrix} ) ↦ ( \begin{matrix}1 \\ χ_{5} \\ χ_{3} \\ χ′_{3} \end{matrix} ) .

The case p = 3. The normalizer N_{G}(S_{3}) of a Sylow 3–subgroup S_{3} of G is isomorphic to the symmetric

group 𝔖_{3}. The principal block B_{3}(N_{G}(S_{3})) is its only 3–block.

1.8. Character table of 𝔖_{3}

(1)  (2)  (3)
1  1  1  1
β_{1}  1  −1  1
β_{2}  2  0  −1

We denote by I_{3} the map

I_{3} : ( \begin{matrix}1 \\ β_{1} \\ β_{2} \end{matrix} ) ↦ ( \begin{matrix}1 \\ χ_{4} \\ χ_{5} \end{matrix} ) .

The case p = 5.

The normalizer N_{G}(S_{5}) of a Sylow 5–subgroup S_{5} of G is isomorphic to the dihedral group D_{5}. The principal block B_{5}(N_{G}(S_{5})) is its only 5–block. Let us change the sign of certain irreducible characters in its character table.

1.9. Character table of D_{5}

(1)  (2)  (5)  (5′)
1  1  1  1  1
−γ_{1}  −1  1  −1  −1
−γ_{2}  −2  0  (1 + \sqrt{5})/2  (1 − \sqrt{5})/2
−γ′_{2}  −2  0  (1 − \sqrt{5})/2  (1 + \sqrt{5})/2

We denote by I_{5} the map

I_{5} : ( \begin{matrix}1 \\ −γ_{1} \\ −γ_{2} \\ −γ′_{2} \end{matrix} ) ↦ ( \begin{matrix}1 \\ χ_{4} \\ χ_{3} \\ χ′_{3} \end{matrix} ) .

Remark. We can also notice that

• if one changes 5 to −3, then I_{2}(α) takes also on the 3–elements of G the same values as α takes on the 5–elements of N_{G}(S_{2}),

• I_{3}(β) takes also on the 2–elements of G the same values as β takes on the 5–elements of N_{G}(S_{3}),

• I_{5}(γ) takes also on the 2–elements of G the same values as γ takes on the

3–elements of N_{G}(S_{5}), i.e., in other words, if {p, q, r} = {2, 3, 5}, not only do ζ and I_{p}(ζ) take the same values on non trivial p–elements, but (with suitable trick for p = 2) they exchange values on non trivial q–elements and r–elements. This last property will not be explained in what follows.

2. Isotypies

The case of 𝔄_{5} provides particular examples of what should replace the factorization type theorem (see above theorem 1.3 and proposition 1.5) in the case where H is the normalizer of an abelian Sylow p–subgroup.

Notation.

Various class functions.

For R = K or O, we denote by CF(G, R) the R–module of all class functions from G into R. For B a block of G, we denote by CF(B, K) the subspace of CF(G, K) consisting of functions which are linear combination of characters of KB, and we set CF(B, O) := CF(G, O) ∩ CF(B, K) .

We denote by CF_{p′}(G, R) the submodule of CF(G, R) consisting of all class functions on G which vanish outside the set G_{p′} of p′–elements of G. For B a block of G, we set CF_{p′}(B, R) := CF_{p′}(G, R) ∩ CF(B, R) .

Remark. The Brauer character of a kG–module is usually defined as a class function on G_{p′}. Extending it by 0 outside G_{p′} , we shall make here the convention that the Brauer characters are elements of CF_{p′}(G, O) (hence elements of CF_{p′}(B, O) for the characters of B–modules). It results from our convention that the set BrIrr(B) of Brauer irreducible characters of B–modules is an O–basis of CF_{p′}(B, O).

Finally, we denote by CF_{p′}^{pr}(G, O) the dual submodule of CF_{p′}(G, O) in CF_{p′}(G, K), consisting of all elements of CF_{p′}(G, K) whose scalar product with the elements of CF_{p′}(G, O) belongs to O, and we set CF_{p′}^{pr}(B, O) := CF_{p′}^{pr}(G, O) ∩ CF(B, K) . From our previous convention it follows that the set Prim(B) of characters of indecomposable projective OG–modules is an O–basis of CF_{p′}^{pr}(G, O).

Decomposition maps.

For x a p–element of G, we denote by

d_{G}^{x}: CF(G, O) → CF_{p′}(C_{G}(x), O)

the linear map defined by

d_{G}^{x}(χ)(y) := \{ \begin{matrix}χ(xy) if y is a p–regular element of C_{G}(x) \\ 0 if y is a p–singular element of C_{G}(x) . \end{matrix}

It results from Brauer’s second and third main theorems (see for example [Fe]) that the map d_{G}^{x} sends CF(B_{p}(G), O) into CF_{p′}(B_{p}(C_{G}(x)), O), and so induces by restriction a map still denoted by

d_{G}^{x}: CF(B_{p}(G), O) → CF_{p′}(B_{p}(C_{G}(x)), O) .

Isotypies.

From now on, the following hypothesis and notation will be in force : We denote by G a finite group whose Sylow p–subgroups are abelian. We denote by S_{p} one of the Sylow p–subgroups, and we set H := N_{G}(S_{p}).

The following definition is a slight modification of the analogous definition given in [Br1] (see Remarque 2 following definition 4.6 in loc. cit.).

Remark. A more general definition is available for non–principal blocks with abelian defect groups. Its statement requires the use of the “local structure” associated with a block (see [AlBr] or [Br4]). We refer the reader to [Br1] for details.

2.1. Definition. An isotypy I between B_{p}(G) and B_{p}(H) is the datum, for every p-subgroup P of S_{p}, of a bijective isometry

I(P): ℤIrr(B_{p}(C_{H}(P)))⟶^{∼}ℤIrr(B_{p}(C_{G}(P)))

such that the following conditions are fullfilled : (Equi) (Equivariance) For all h ∈ H, we have I(P)^{h} = I(P^{h}) . (Com) (Compatibility condition) For every subgroup P of S_{p} and every x ∈ S_{p}, we still

denote by

I(P): CF(B_{p}(C_{H}(P)), K): ⟶^{∼}CF(B_{p}(C_{G}(P)), K)

the bijective isometry defined by linear extension of I(P). The following diagram is commutative :

CF(B_{p}(C_{H}(P)), K) ⟶^{I(P)} CF(B_{p}(C_{G}(P)), K)
d_{C_{H} (P)}^{x}↓  ↓^{d_{C_{G}(P)}^{x}}
CF(B_{p}(C_{H}(P ⟨x⟩)), K) ⟶^{I(P ⟨x⟩)} CF(B_{p}(C_{G}(P ⟨x⟩)), K)

(Triv) I(S_{p}) is the identity map. Moreover, we say that the isotypy I is normalized if I(P)(1_{C_{H}(P)}) = 1_{C_{G}(P)} for all P ⊆ S_{p}.

Let us list some of the straighforward properties of an isotypy. In what follows, for every subgroup P of S_{p} we denote by R(P) the inverse map of I(P). We set I(1) = I({1}) and R(1) = R({1}).

2.2. Proposition. (Loc) (Local isotypies) Let P be any subgroup of S_{p}. Set I_{P} (Q) := I(P Q) . The

collection of maps (I_{P} (Q))_{Q⊆S_{p}} defines an isotypy between B_{p}(C_{H}(P)) and

B_{p}(C_{G}(P)) .

(Int) (Integrality property) By restriction, the map I(1) induces bijective isometries

I(1): CF(B_{p}(H), O) ⟶^{∼} CF(B_{p}(G), O)
I(1): CF_{p′}(B_{p}(H), O) ⟶^{∼} CF_{p′}(B_{p}(G), O)
I(1): CF_{p′}^{pr}(B_{p}(H), O) ⟶^{∼} CF_{p′}^{pr}(B_{p}(G), O)

(Loc) is obvious. We give a proof of (Int). By 2.1, (Com), applied with x = 1, we see first that I(1) sends CF_{p′}(B_{p}(H), K) into CF_{p′}(B_{p}(G), K). Moreover, since, for any finite group G, the ordinary decomposition map d_{G}^{1} : ℤIrr(B_{p}(G)) → ℤBrIrr(B_{p}(G)) is onto, and since BrIrr(B_{p}(G)) is an O–basis of CF_{p′}(B_{p}(G), O), we see that I(1) and R(1) define inverse isometries between CF_{p′}(B_{p}(H)), O) and CF_{p′}(B_{p}(G), O). It then follows by adjunction that they induce inverse isometries between CF_{p′}^{pr}(B_{p}(H), O) and CF_{p′}^{pr}(B_{p}(G), O).

To check that I(1) and R(1) induce inverse isomorphisms between CF(B_{p}(H), O) and CF(B_{p}(G), O), it suffices to check that the image of CF(B_{p}(H), O) under I(1) is contained in CF(B_{p}(G), O), i.e.,that, for ζ ∈ CF(B_{p}(H), O) and g ∈ G, we have I(1)(ζ)(g) ∈ O. Let x be the p–component of g and let x′ be its p′–component. We have I(1)(ζ)(g) = (I(⟨x⟩)(d_{H}^{x}(ζ))) (x′) and the result follows from the fact that I(⟨x⟩) sends CF_{p′}(C_{H}(x), O) into CF_{p′}(C_{G}(x), O), by the “local isotypies” property (Loc) and what preceds. □

Remark.

The integrality properties (Int) show that the map I(1) is a “perfect isometry” as defined in [Br1].

It follows in particular (see [Br1] or [Br5]) that an isotypy induces an isomorphism between the associated “Cartan–decomposition” triangles between Grothendieck groups (see [Se] or [Br1] for the definitions of the triangles).

[figure: ℤIrr(B_{p}(G)); d_{G}^{1}; ℤBrIrr(B_{p}(G)); ℤPrim(B_{p}(G)); ^{t}d_{G}^{1}; c_{G}; ℤIrr(B_{p}(H)); d_{H}^{1}; ℤBrIrr(B_{p}(H)); ℤPrim(B_{p}(H)); ^{t}d_{H}^{1}; c_{H}]

The character of an isotypy.

Any linear map F: ℤIrr(H) → ℤIrr(G) between the character groups of H and G defines a character μ_{F} of H × G by the formula

μ_{F} := ∑_{ζ∈Irr(H)} ζ ⊗ F (ζ) .

Let I = (I(P))_{P ⊆S_{p}} be an isotypy between B_{p}(G) and B_{p}(H). We set μ_{P} := μ_{I(P)} . Then the defining properties of an isotypy (cf. 2.1) translate as follows :

(Equi)  For all h ∈ H, we have μ_{P}^{h} = μ_{P^{h}} .

(Com)  μ_{P} (xx′, yy′) = \{ \begin{matrix}0 & if x and y are not conjugate in G , \\ μ_{P ⟨x⟩}(x′, y′) & if x = y . \end{matrix}

The following statement describes the case of a p–group. If G is an (abelian) p–group, then H = G, and B_{p}(G) = OG.

2.3. Lemma. Let G be an abelian p–group. Let I = (I(P))_{P ⊆G} be a “self–isotypy” of OG. If p = 2, assume moreover that I is normalized. Then I(P) is the identity for all p–subgroups P of G.

Proof of 2.3. By definition of an isotypy (property (Triv)), we know that I(G) is the identity. We prove by induction on |G : P | that I(P) = Id for any subgroup P of G. Let P be a proper subgroup of G. By the induction hypothesis, we may assume that I(P′) = Id whenever P′ is a subgroup of G which strictly contains P, and we must prove that I(P) = Id. For x ∈ G, the map d_{G}^{x} is identified with the map χ ↦ χ(x). For χ ∈ Irr(G) we have I(P)(χ)(x) = d_{G}^{x}(I(P)(χ)) = I(P ⟨x⟩)(d_{G}^{x}(χ)) and so I(P)(χ)(x) = χ(x) for all x ∉ P. Now let y ∈ P. For x ∉ P, we have xy ∉ P, so I(P)(χ)(xy) = χ(xy). Set I(P)(χ) = ε_{χ}χ′ where ε_{χ} = ±1 and χ′ ∈ Irr(G). It follows that

χ′(z) = \{ \begin{matrix}ε_{χ}χ(z) & for z ∉ P \\ χ(z) & for z ∈ P . \end{matrix}

For p odd, this implies that ε_{χ} = 1, and so I(P) = Id.

Assume p = 2. Since I is normalized, I(P) fixes the trivial character, and so for x ∈ P we have I(P)(χ)(x) = I(P ⟨x⟩)(d_{G}^{x}(χ)) = I(P)(d_{G}^{x}(χ)) = d_{G}^{x}(χ) = χ(x) which also shows that ε_{χ} = 1 and that I(P) = Id. □

Remark. If G = ℤ/2ℤ and if Irr(G) = {1, σ}, then I := {I(1), I(G)} where I(G) = Id and

I(1): \{ \begin{matrix}1 & ↦ −σ \\ σ & ↦ −1 \end{matrix}

is a non trivial self–isotypy.

A conjecture.

The following conjecture is a particular case of a more general conjecture concerning blocks with abelian defect groups (see [Br1]).

2.4. Conjecture. Let G be a finite group whose Sylow p–subgroups are abelian. Let H be the normalizer of a Sylow p-subgroup. Then there is a normalized isotypy between B_{p}(G) and B_{p}(H).

The preceding conjecture is true if G is p–solvable, by 1.4 and 1.5, (2), above.

It has also been checked in the following cases.

• G is a symmetric group (Rouquier, [Rou1]), or an alternating group (Fong),

• G is a sporadic non abelian simple group (Rouquier, [Rou1]),

• G is a “finite reductive group” in non-describing characteristic (Broué–Malle– Michel, [BMM] and [BrMi]),

• p = 2 — and G is any finite group with abelian Sylow 2–subgroups (Fong–Harris, [FoHa]).

3. Rickard equivalences

We shall explain now why the existence of an isotypy between the principal blocks of two finite groups must be the “shadow” of a much deeper connection between the two blocks, which we call a “Rickard equivalence”. p–permutation modules and Rickard complexes.

p–permutation modules and the Brauer functor.

Let R denote either O or k. We call p–permutation RG–modules the summands of the permutation G–modules over R. The following characterization of p–permutation RG–modules is well known (see for example [Br3]).

3.1. Proposition. The p–permutation RG–modules are the modules which, once restricted to a Sylow p–subgroup of G, are permutation modules.

Let us denote by _{RG}perm_{p} the category of all p–permutation RG–modules. For P a p–subgroup of G, we set \overline{N}_{G}(P) := N_{G}(P)/P .

For Ω a finite G–set, we denote by Ω^{P} the set of fixed points of Ω under P, viewed as a \overline{N}_{G}(P)–set.

3.2. Proposition. There is a functor

Br_{P} : _{OG}perm_{p} → _{k\overline{N}_{G}(P)}perm_{p}

which “induces” the “fixed points” functor, i.e., which is such that the diagram of natural transformations

Gset ⟶^{·^{P}}  _{\overline{N}_{G}(P)}set
↓  ↓
OGperm_{p} ⟶^{Br_{P}} _{k\overline{N}_{G}(P)}perm_{p}

is commutative.

Sketch of proof of 3.2. For X a p–permutation OG–module and Q a p–subgroup of G, we define

Tr_{Q}^{P}: X^{Q} → X^{P} by

Tr_{Q}^{P}(x) := ∑_{g∈P/Q} g(x) .

We set

Br_{P} (X) := (X/𝔭X)^{P} / ∑_{Q⊊P} Tr_{Q}^{P}((X/𝔭X)^{Q}) .

□

For V any kG–module, and g a p′–element of G, we denote by Brtr(g; V) the value at g of the Brauer character of V. Recall that we view the Brauer character Brtr(·; V) as a class function on G vanishing outside the set G_{p′} of p–regular elements of G. The following proposition generalizes to p–permutation modules a result which is well known for actual permutation modules.

3.3. Proposition. Let X be a p–permutation OG–module. If g = g_{p}g_{p′} where g_{p} is a p–element, g_{p′} is p-regular and g_{p}g_{p′} = g_{p′}g_{p}, then

tr(g; X) = Brtr(g_{p′} ; Br_{⟨g_{p}⟩}(X)) .

Rickard complexes.

Let us start with some notation related to complexes of modules.

Let Γ := (⋯ → 0 → Γ^{m}⟶^{d^{m}}Γ^{m+1} ⟶^{d^{m+1}} ⋯ ⟶^{d^{m+a−1}}Γ^{m+a} → 0 → ⋯) be a complex of

modules on some O–algebra. The O–dual of Γ is by definition the complex Γ^{∗} :=

(⋯ → 0 → Hom_{O}(Γ^{m+a}, O) ⟶^{^{t}d^{m+a−1}} ⋯ ⟶^{^{t}d^{m}}Hom_{O}(Γ^{m}, O) → 0 → ⋯) .

From now on, the following hypothesis and notation will be in force :

We denote by G a finite group whose Sylow p–subgroups are abelian. We denote by S_{p} one of the Sylow p–subgroups, and we set H := N_{G}(S_{p}).

3.4. Definition. A Rickard complex for the principal blocks B_{p}(G) and B_{p}(H) is a bounded complex of (B_{p}(G), B_{p}(H))–bimodules

Γ := (⋯ → 0 → Γ^{m}⟶^{d^{m}}Γ^{m+1} ⟶^{d^{m+1}} ⋯ ⟶^{d^{m+a−1}}Γ^{m+a} → 0 → ⋯)

with the following properties :

(1) Each constituant Γ^{n} of Γ, viewed as an O[G × H]–module, is a p–permutation module with vertex contained in Δ_{G×H^{o}}(S_{p}) (where Δ_{G×H^{o}}: S_{p} → G × H is defined by Δ_{G×H^{o}}(x) := (x, x^{−1})).

(2) We have homotopy equivalences :

Γ ⊗_{OH} Γ^{∗} ≃ B_{p}(G) as complexes of (B_{p}(G), B_{p}(G))–bimodules,
Γ^{∗} ⊗_{OG} Γ ≃ B_{p}(H) as complexes of (B_{p}(H), B_{p}(H))–bimodules.

One of the main properties of Rickard complexes is that they automatically define Rickard complexes at the “local level” as well, as shown by the following result.

3.5. Theorem. (J. Rickard)

Let Γ be a Rickard complex for B_{p}(G) and B_{p}(H). Then, for every subgroup P of S_{p}, there is a finite complex Γ_{P} of (B_{p}(C_{G}(P)), B_{p}(C_{H}(P)))–bimodules, unique up to isomorphism, such that

(1) Γ_{P} is a Rickard complex for B_{p}(C_{G}(P)) and B_{p}(C_{H}(P)),

(2) we have

Br_{Δ_{G×H^{o}} (P)}(Γ) = k ⊗_{O} Γ_{P} .

Rickard complexes and derived equivalences.

For A an O–algebra (finitely generated as an O-module), we denote by D^{b}(A) the derived bounded category of the module category _{A}mod, i.e., the triangulated category whose

• objects are the complexes

X := (⋯ → X^{n}⟶^{d^{n}}X^{n+1}⟶^{d^{n+1}} ⋯ ⟶^{d^{a−1}}X^{a} → 0 → ⋯)

of finitely generated projective A–modules, bounded on the right, and exact almost everywhere,

• morphisms are chain maps modulo homotopy.

If Γ is a Rickard complex for B_{p}(G) and B_{p}(H), it is easy to see that the functor Y ↦ Γ ⊗_{B_{p}(H)} Y defines an equivalence of triangulated categories from D^{b}(B_{p}(H)) to

D^{b}(B_{p}(G)) (see for example [Br5] for more details).

Thus the datum of a Rickard complex for B_{p}(G) and B_{p}(H) induces a consistent family of derived equivalences between B_{p}(C_{G}(P)) and B_{p}(C_{H}(P)), where P runs over the set of subgroups of S_{p}.

Rickard complexes and isotypies.

• Let

Γ := (⋯ → 0 → Γ^{m}⟶^{d^{m}}Γ^{m+1} ⟶^{d^{m+1}} ⋯ ⟶^{d^{m+a−1}}Γ^{m+a} → 0 → ⋯)

be a Rickard complex for B_{p}(G) and B_{p}(H). We denote by μ_{Γ} the character of Γ as a complex of (OG, OH)–bimodules, i.e., μ_{Γ} := ∑_{n}(−1)^{n}tr(·; Γ^{n}) .

• For every subgroup P of S_{p}, let Γ_{P} be the complex of (B_{p}(C_{G}(P)), B_{p}(C_{H}(P)))– bimodules defined as in 3.5 above. We denote by μ_{Γ_{P}} the character of Γ_{P} as a complex of (OC_{G}(P), C_{H}(P))–bimodules.

The following result is a consequence of the definition of a Rickard complex, of 3.3 and of 3.5. It shows that a Rickard complex for B_{p}(G) and B_{p}(H) provides a natural isotypy between B_{p}(G) and B_{p}(H).

3.6. Theorem. There is an isotypy I = (I(P))_{P ⊆S_{p}} such that, for each subgroup P of S_{p}, we have

μ_{Γ_{P}} = ∑_{ζ∈Irr(B_{p}(C_{H}(P)))} ζ ⊗ I(P)(ζ) .

A conjecture.

The following conjecture (one of J. Rickard and the author’s dreams) makes more precise earlier conjectures about derived equivalences between blocks (see [Br1] and [Br5]).

3.7. Conjecture. Let G be a finite group with abelian Sylow p–subgroups, and let H be the normalizer of one of the Sylow p–subgroups. Then there exists a Rickard complex for B_{p}(G) and B_{p}(H).

It follows from 3.6 that the preceding conjecture implies the conjecture 2.4.

Notice that conjecture 3.7 holds if G is p–solvable by 1.4. Indeed, in this case the (B_{p}(G), B_{p}(H))–bimodule O(G/O_{p′}G) is a Rickard complex for B_{p}(G) and B_{p}(H) (in this case, the algebras B_{p}(G) and B_{p}(H) are actually isomorphic).

Conjecture 3.7 is known to hold only in a few cases if G is not p–solvable.

• Some examples are provided below for the case G = 𝔄_{5}, and others are provided in §4 for the case where G is a finite reductive group.

• It follows from recent work of Rouquier ([Rou2]) that conjecture 3.7 holds when S_{p} is cyclic.

The case of G = 𝔄_{5}.

The case p = 2.

The following explanation of the isotypy I_{2} described in §1 is due to Rickard (cf. [Ri4]).

We set H := N_{G}(S_{2}), where S_{2} denote a Sylow 2–subgroup of G.

We view B_{2}(G) as a (B_{2}(G), B_{2}(H))–bimodule, where B_{2}(G) acts by left multiplication, while B_{2}(H) acts by right multiplication.

Let us denote by IB_{2}(G) the kernel of the augmentation map B_{2}(G) → O. Thus IB_{2}(G) is a (B_{2}(G), B_{2}(H))–sub-bimodule of B_{2}(G). Let C denote a projective cover of the bimodule IB_{2}(G).

\xymatrix{ & C \ar[d] \ar[dr] \\ \{0\} \ar[r] & IB_{2}(G) \ar[r] \ar[d] & B_{2}(G) \ar[r] & \{0\} \\  & \{0\}}

We denote by

Γ_{2} := ({0} → C → B_{2}(G) → {0})

the complex of (B_{2}(G), B_{2}(H))–bimodules defined by the preceding diagonal arrow, where B_{2}(G) is in degree 0 (and C in degree −1).

We denote by KΓ_{2} the complex of (KB_{2}(G), KB_{2}(H))–bimodules deduced by extension of scalars up to K. Let H^{0}(KΓ_{2}) and H^{−1}(KΓ_{2}) be the corresponding homology groups, viewed as (KG, KH)–bimodules.

It is clear that H^{0}(KΓ_{2}) is the trivial (KG, KH)–bimodule, hence its character is 1 ⊗ 1.

3.8. Theorem. (J. Rickard)

(1) The character of H^{−1}(KΓ_{2}) (with suitables choices of \sqrt{5} and \sqrt{−3} in the field

K) is

(χ_{5} ⊗ α_{3}) + (χ_{3} ⊗ α_{1}) + (χ′_{3} ⊗ α′_{1}) .

In particular the character of ∑_{n}(−1)^{n}H^{n}(KΓ_{2}) is

(1 ⊗ 1) − ((χ_{5} ⊗ α_{3}) + (χ_{3} ⊗ α_{1}) + (χ′_{3} ⊗ α′_{1})) .

(2) We have homotopy equivalences :

Γ_{2} ⊗_{OH} Γ_{2}^{∗} ≃ B_{2}(G) as complexes of (B_{2}(G), B_{2}(G))–bimodules,
Γ_{2}^{∗} ⊗_{OG} Γ_{2} ≃ B_{2}(H) as complexes of (B_{2}(H), B_{2}(H))–bimodules.

Remark. The above theorem 3.8 may be viewed now as a particular consequence of Rouquier’s recent theorem (cf. [Rou2]) which provides Rickard complexes from certain stable equivalences.

The case p = 3. Now we view G as SL_{2}(4). We then denote by T the group of diagonal matrices in G (the split torus, which is also a Sylow 3–subgroup of G), and by U the Sylow 2–subgroup of G, consisting of unipotent uppertriangular matrices.

We set H := N_{G}(T), and σ := ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). We have H = T ⋊ ⟨σ⟩ .

Let Γ_{3} denote the free ℤ_{3}–module with basis G/U. Then Γ_{3} is a (ℤ_{3}G, ℤ_{3}T)– bimodule, where G acts by left multiplication while T acts by right mutiplication.

3.9. Theorem.

(1) The (ℤ_{3}G, ℤ_{3}T)–bimodule Γ_{3} extends to a (ℤ_{3}G, ℤ_{3}H)–bimodule, whose character as a (KG, KH)–bimodule is

(1 ⊗ 1) + (χ_{4} ⊗ β_{1}) + (χ_{5} ⊗ β_{2}) .

(2) We have isomorphisms :

Γ_{3} ⊗_{OH} Γ_{3}^{∗} ≃ B_{3}(G) as (B_{3}(G), B_{3}(G))–bimodules,
Γ_{3}^{∗} ⊗_{OG} Γ_{3} ≃ B_{3}(H) as (B_{3}(H), B_{3}(H))–bimodules.

Remark. The previous example is a particular case of a much more general situation, as will be shown in §4.

The case p = 5.

Now we set H := N_{G}(S_{5}), where S_{5} is a Sylow 5–subgroup of G. Since S_{5} is cyclic, we can apply Rouquier’s constructions as in [Rou2]. The Brauer trees of the blocks B_{5}(G) and B_{5}(H) are respectively

(for G)  [figure: 1; P_{1}; χ_{4}; P_{2}; χ_{ex}]

(for H)  [figure: 1; Q_{1}; γ_{ex}; Q_{2}; γ_{1}]

where the exceptional vertices correspond to characters

χ_{ex} := χ_{3} + χ′_{3}
γ_{ex} := γ_{2} + γ′_{2} .

In other words, we have minimal projective resolutions of the trivial representations, periodic of period 4, of the following shapes :

⋯ → P_{1} → P_{2} → P_{2} → P_{1} → O → {0}
⋯ → Q_{1} → Q_{2} → Q_{2} → Q_{1} → O → {0} .

The (B_{5}(G), B_{5}(H))–bimodule B_{5}(G) is indecomposable and its projective cover has the shape (cf. [Rou2])

(P_{1} ⊗ Hom_{O}(Q_{1}, O)) ⊕ (P_{2} ⊗ Hom_{O}(Q_{2}, O)) ⟶^{π}B_{5}(G) .

Following [Rou2], we then define

Γ_{5} := ({0} → P_{2} ⊗ Hom_{O}(Q_{2}, O)⟶^{π}B_{5}(G) → {0}) .

3.10. Theorem. (R. Rouquier)

(1) The character of H^{−1}(KΓ_{5}) is

(χ_{4} ⊗ γ_{1}) + (χ_{3} ⊗ γ_{2}) + (χ′_{3} ⊗ γ′_{2}) .

In particular the character of ∑_{n}(−1)^{n}H^{n}(KΓ_{5}) is

(1 ⊗ 1) − ((χ_{4} ⊗ γ_{1}) + (χ_{3} ⊗ γ_{2}) + (χ′_{3} ⊗ γ′_{2})) .

(2) We have homotopy equivalences :

Γ_{5} ⊗_{OH} Γ_{5}^{∗} ≃ B_{5}(G) as complexes of (B_{5}(G), B_{5}(G))–bimodules,
Γ_{5}^{∗} ⊗_{OG} Γ_{5} ≃ B_{5}(H) as complexes of (B_{5}(H), B_{5}(H))–bimodules.

4. The case of the finite reductive groups

In the case where G is a “finite reductive group”, the conjecture 3.7 can be made more precise and closely linked with the underlying algebraic geometry (for more details, see [BrMa]).

In this paragraph, we change our notation to fit with the usual notation of finite reductive groups : our prime p (the characteristic of our field k := O/𝔭) is now denoted by ℓ, and q denotes a power of another prime p ≠ ℓ.

From now on, we denote by G be a connected reductive algebraic group over \overline{𝔽}_{q}, endowed with a Frobenius endomorphism F which defines a rational structure on 𝔽_{q}. The finite group G^{F} of fixed points of G under F is called a finite reductive group.

The Deligne–Lusztig variety and its ℓ–adic cohomology.

The Deligne–Lusztig variety.

Let P be a parabolic subgroup of G, with unipotent radical U, and with F –stable Levi subgroup L.

We denote by Y(U) the associated Deligne–Lusztig variety defined (cf. [DeLu] and [Lu]) by

Y(U) := {g(U ∩ F (U)) ∈ G/U ∩ F (U) ; g^{−1}F (g) ∈ F (U)} .

It is clear that G^{F} acts on Y(U) by left multiplication while L^{F} acts on Y(U) by right multiplication.

It is known (cf. [Lu]) that Y(U) is an L^{F} –torsor on a variety X(U), which is smooth of pure dimension equal to dim(U/U ∩ F (U)), and which is affine (at least if q is large enough). In particular X(U) is endowed with a left action of G^{F}. If R is a commutative ring, the image of the constant sheaf R on Y(U) through the finite morphism π: Y(U) → X(U) is a locally constant constant sheaf π_{∗}(R) on X(U). We denote this sheaf by F_{RL^{F}}.

A consequence of yet another theorem of J. Rickard.

From now on, we denote by ℓ a prime number which does not divide q, and which is good for G.

The following theorem is a consequence of the main result of [Ri5] (for a “character theoretic approach” of this result, see [Br1], §2.A).

4.1. Theorem. (J. Rickard) There exists a bounded complex

Λ_{c}(X(U), F_{ℤ_{ℓ}L^{F}}) = (⋯ → 0 → Λ^{m}⟶^{d^{m}}Λ^{m+1} ⟶^{d^{m+1}} ⋯ ⟶^{d^{m+a−1}}Γ^{m+a} → 0 → ⋯)

of (ℤ_{ℓ}G^{F} , ℤ_{ℓ}L^{F})–bimodules, with the following properties :

(1) For each positive integer n, (ℤ_{ℓ}/ℓ^{n}ℤ_{ℓ}) ⊗ Λ_{c}(X(U), F_{ℤ_{ℓ}L^{F}}) is a representative, in the derived bounded category of ((ℤ_{ℓ}/ℓ^{n}ℤ_{ℓ})G^{F} , (ℤ_{ℓ}/ℓ^{n}ℤ_{ℓ})L^{F})–bimodules, of the “ℓ–adic cohomology complex” RΓ_{c}(X(U), F_{(ℤ_{ℓ}/ℓ^{n}ℤ_{ℓ})L^{F}}).

(2) For each integer n, the ℤ_{ℓ}[G^{F} × L^{F}]–module Λ^{n} is an ℓ–permutation module, such that each of its indecomposable constituant has a vertex contained in Δ_{G^{F} ×(L^{F})^{o}}(L^{F}).

If O is the ring of integers of a finite extension K of ℚ_{ℓ}, we set

Λ_{c}(X(U), F_{OL^{F}}) := O ⊗_{ℤ_{ℓ}} Λ_{c}(X(U), F_{ℤ_{ℓ}L^{F}}) .

A conjecture.

Notation.

From now on, we assume that ℓ is a prime number, ℓ ≠ p, which is good for G, and such that the Sylow ℓ–subgroups of G^{F} are abelian.

• Let O be the ring of integers of a finite unramified extension k of the field of ℓ–adic numbers ℚ_{ℓ}, with residue field k, such that the finite group algebra kG^{F} is split.

• Let e be the principal block idempotent of OG^{F}, so that OG^{F}e is the principal block B_{ℓ}(G^{F}) of OG^{F}.

• Let S be a Sylow ℓ–subgroup of G^{F} and let L := C_{G}(S). The group L is a rational Levi subgroup of G.

We have N_{G^{F}} (S) = N_{G^{F}} (L). The group S is a Sylow ℓ–subgroup of Z(L)^{F}, and ℓ does not divide |N_{G^{F}} (L)/L^{F} |.

Let f be the principal block idempotent of OL^{F}, so that OL^{F}f is the principal block B_{ℓ}(L^{F}) of OL^{F}.

4.2. Conjecture. There exists a parabolic subgroup of G with unipotent radical U and Levi complement L, such that (C1) the idempotent e acts as the identity on the complex Λ_{c}(X(U), F_{OL^{F}}).f, (C2) the structure of complex of (OG^{F}e, OL^{F}f)–bimodules of Λ_{c}(X(U), F_{OL^{F}}).f ex-

tends to a structure of complex of (OG^{F}e, ON_{G^{F}} (L)f)–bimodules, (C3) we have homotopy equivalences :

Λ_{c}(X(U), F_{OL^{F}}).f ⊗_{ON_{G^{F}} (L)f} f.Λ_{c}(X(U), F_{OL^{F}})^{∗} ≃ OG^{F}e
as complexes of (OG^{F}e, OG^{F}e)–bimodules,
f.Λ_{c}(X(U), F_{OL^{F}})^{∗} ⊗_{OG^{F} e} Λ_{c}(X(U), F_{OL^{F}}).f ≃ ON_{G^{F}} (L)f
as complexes of (ON_{G^{F}} (L)f, ON_{G^{F}} (L)f)–bimodules.

By 4.1, one sees that if the above conjecture is true, the complex Λ_{c}(X(U), F_{OL^{F}}) is indeed a Rickard complex for OG^{F}e and ON_{G^{F}} (L)f.

Although some evidence in favour of conjecture 4.2 is indeed available (see [BrMa]), it is actually known to be true in very few cases.

The particular case where ℓ divides (q − 1) (which had been conjectured and almost proved by Hiß) may be deduced from some results of Puig ([Pu]).

4.3. Theorem. Assume that ℓ divides q − 1 and does not divide the order of the Weyl group of G.

(1) We have L = T, a quasi–split maximal torus of G. For P = B, a rational Borel subgroup of G containing T with unipotent radical U, we have

Λ_{c}(X(U), F_{OL^{F}}) ≃ O[G^{F} /U^{F}] .

(2) (L. Puig) We have isomorphims :

O[G^{F} /U^{F}].f ⊗_{ON_{G^{F}} (T)f} f.O[U^{F} \G^{F}] ≃ OG^{F}e
as (OG^{F}e, OG^{F}e)–bimodules,
f.O[U^{F} \G^{F}] ⊗_{OG^{F} e} O[G^{F} /U^{F}].f ≃ ON_{G^{F}} (T)f
as (ON_{G^{F}} (T)f, ON_{G^{F}} (T)f)–bimodules,

providing a Morita equivalence between OG^{F}e and ON_{G^{F}} (T)f.

Remark. The case where G^{F} = SL_{2}(4) and ℓ = 3 (see theorem 3.9 above) is a particular case of the preceding theorem.

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L.M.E.N.S.–D.M.I. (C.N.R.S., U.A. 762), 45 rue d’Ulm, F–75005 Paris, France E-mail address: Michel.Broue@ens.fr